First off, I do understand that there's a difference and that the two words don't necessarily mean the same (i.e. desert - 'desert-like', bare of vegetation; deserted - 'desolate, uninhabited'). But the dictionaries give another definition for 'desert', which made me think that there's an overlap.
From M-W dictionary:
Main Entry: 2des·ert
Pronunciation: \ˈde-zərt\
Function: adjective
Date: 13th century
1 : desolate and sparsely occupied or unoccupied <a desert island>
Is this not an adjective?
Yes, it is. The index word 'desert' is defined as being an adjective; 'desolate', 'occupied' and 'unoccupied' are adjectives, and 'sparsely' is an adverb.
I take your word for it that 'desert wood' is not possible.
Do you understand why?
But I don't understand your explanation that the defintion above is somehow different from that of simply 'desolate'.
Well, on the face of it, "desolate" and "desolate and sparsely occupied or unoccupied" are different, unless you claim that "sparsely occupied or unoccupied" is strictly synonymous with 'desolate'. But if this were the case, why has the dictionary committed this redundancy?
In general, I think you'd accept that A + B does not equal A, unless B is zero, or of no consequence in the context.
All I'm saying is that "desolate and sparsely occupied or unoccupied" obviously had different connotations to the lexicographer than "desolate" by itself did. I don't know what difference the lexicographer had in mind, but perhaps s/he was attempting to clarify that other meanings of 'desolate' don't mean 'desert'(adj). For example, a women who has lost a child can be desolate. But, since she is not "sparsely occupied or unoccupied", she cannot be described as being "a desert woman".
Could you please enlighten me?
I can do my best.